6th Grade · 6.RP.A.1

Ratios in 6th grade

Ratios compare two amounts, such as 4 bananas to 9 apples. In 6th grade, your child learns to write a ratio in different ways and to describe it with ratio language, such as "for every" and "to". Rates and percents later in the year build on this idea.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

What it means, in plain words

A ratio compares two quantities. If a team won 10 games and lost 6, the ratio of wins to losses is 10 to 6. Your child can write it as $10:6$ or as "10 to 6". The order matters. The ratio of losses to wins is $6:10$ , which describes the same team the other way around.

Ratio language is the set of phrases used to describe a ratio. "For every 10 wins there were 6 losses" and "there were 10 wins for each 6 losses" both mean $10:6$ . A ratio can compare two parts, like wins and losses. It can also compare a part with the whole. The team played 16 games, so the ratio of wins to games played is $10:16$ .

Ratios compare by groups and multiplication, not by subtraction. "There were 4 more wins than losses" is a true sentence, but it is a different comparison. A ratio of $1:3$ means there are 3 times as many of the second thing as of the first.

Your child should be able to:

Write a ratio as $a:b$ , as "a to b" and in words
Describe a ratio with ratio language such as "for every" and "for each"
Keep the two amounts in the order the question asks for
Tell a ratio of two parts from a ratio of a part to the whole

Example

A fruit bowl has 4 bananas and 9 apples. What is the ratio of bananas to apples, and what is the ratio of apples to all the fruit?

Bananas come first, so the ratio of bananas to apples is $4:9$ , or 4 to 9.
In ratio language: for every 4 bananas there are 9 apples.
All the fruit is $4 + 9 = 13$ pieces.
The ratio of apples to all the fruit is $9:13$ .

Example

In a tile pattern, every row has 1 blue tile and 3 white tiles. What does the ratio of white tiles to blue tiles mean?

The ratio of white tiles to blue tiles is $3:1$ .
In ratio language: for each blue tile there are 3 white tiles.
So there are 3 times as many white tiles as blue tiles. A wall with 5 blue tiles has $5 \times 3 = 15$ white tiles.

Where children usually get stuck

Asked for the ratio of bananas to apples, your child writes $9:4$ .
The child writes the bigger number first or follows the order of a picture, not the order of the words in the question.
What helps. Have your child write the words under the numbers: 4 bananas : 9 apples. The words must come in the same order as in the question.
Your child says that bananas are $\frac{4}{9}$ of the fruit in the bowl.
The ratio $4:9$ looks like the fraction $\frac{4}{9}$ . But 9 counts only the apples, and a fraction of the fruit needs the whole bowl, 13 pieces, on the bottom.
What helps. Ask how many pieces of fruit there are in all. Bananas are 4 out of 13 pieces, or $\frac{4}{13}$ of the fruit.
For a pattern of 1 blue tile to 3 white tiles, your child says a wall with 5 blue tiles has 7 white tiles.
The child sees 2 more white tiles than blue ones and keeps that difference. A ratio compares by multiplying, not by adding the same amount.
What helps. Draw the tiles as groups, where each blue tile comes with its own 3 white tiles. Five groups make 15 white tiles.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
For every 2 cats, there are 3 dogs. Which ratio matches the sentence? Keep the quantities in the same order as the sentence.
A
2:3
B
3:2
C
2:4
D
3:3
Show the answer and how to solve it
The answer is 2:3 because the sentence says 2 cats first and 3 dogs second.
1. 1Read the sentence in order. The first quantity is 2 cats, and the second quantity is 3 dogs.
2. 2Write the ratio in that same order.
3. 3Check that both numbers match the sentence exactly. Changing either number by even 1 gives the wrong ratio.
Common mistake. A common mistake is choosing 3:2 because it reverses the order. Another mistake is writing 2:4 or 3:3 by changing one of the numbers by 1.
The idea underneath. A ratio written from a sentence keeps the quantities in the exact order they are said.
2On level
The ratio of red stickers to blue stickers is shown below.
$\displaystyle 5:2$
Select the sentence that matches the ratio.
A
For every 2 red stickers, there are 5 blue stickers.
B
For every 5 red stickers, there are 2 blue stickers.
C
For every 5 blue stickers, there are 2 red stickers.
D
For every 5 red stickers, there are 7 blue stickers.
Show the answer and how to solve it
The correct sentence is For every 5 red stickers, there are 2 blue stickers because the ratio keeps the same order as the quantities named.
1. 1The prompt says the ratio compares red stickers to blue stickers, in that order.
2. 2In 5:2, the first number 5 goes with red stickers and the second number 2 goes with blue stickers.
3. 3So the matching sentence is: For every 5 red stickers, there are 2 blue stickers.
Common mistake. A common mistake is reversing the order and reading 5:2 as 2 red stickers to 5 blue stickers. In a ratio, the first number always matches the first quantity named.
The idea underneath. Ratio language keeps order: the first named quantity matches the first number, and the second named quantity matches the second number.
3On level
Alex tried to rewrite a ratio in the requested form. Look at the work below.
$\displaystyle \begin{array}{l} \text{Original ratio: } 4:7 \\[0.5em] \text{Requested form: fraction} \\[0.5em] \text{Alex's rewrite: } \frac{4}{7} \end{array}$
Select the best description of Alex's work.
A
reversed the order
B
changed the numbers
C
used the wrong form
D
no mistake
Show the answer and how to solve it
The correct answer is no mistake because Alex kept the numbers 4 and 7 in the correct order and used fraction form, writing exactly $\frac{4}{7}$ .
1. 1Read the three labeled lines: the original ratio is 4:7, the requested form is fraction, and Alex wrote $\frac{4}{7}$ .
2. 2To write 4:7 as a fraction, keep the same order and put 4 over 7.
3. 3Check all three conditions: same numbers (4 and 7 ✓), same order (4 first, 7 second ✓), correct form (fraction ✓). All three checks pass.
4. 4All three checks pass, so the best description is no mistake.
Common mistake. A common mistake is choosing changed the numbers because $\frac{4}{7}$ looks different from 4:7. The notation changed, but both numbers stayed exactly the same.
The idea underneath. When you convert a ratio form, only the notation changes; the two numbers and their order stay the same.
4Stretch
At the library sale, there are 11 fiction books for every 7 nonfiction books.
Enter the ratio of nonfiction books to fiction books in a:b form.
Loading...
Show the answer and how to solve it
Answer. 7:11
The answer is 7:11 because the instruction asks for nonfiction books first, then fiction books, giving 7:11.
1. 1The ratio compares nonfiction books and fiction books.
2. 2The sentence gives 11 fiction books and 7 nonfiction books. There is no extra total here, so both numbers are used.
3. 3The sentence mentions fiction first (11), but the instruction asks for nonfiction to fiction — so the answer order follows the instruction, not the sentence order.
4. 4Write nonfiction first (7), then fiction second (11), as the instruction requires.
Common mistake. A common mistake is writing 11:7 by copying the order from the sentence instead of following the instruction, which asks for nonfiction to fiction.
The idea underneath. Use the numbers from the 'for every' statement. Write them in the order the question asks for, with a colon between them.